A Galactic Plane Defined by the Milky Way HII Region Distribution

Draft version December 7, 2018 Typeset using LATEX twocolumn style in AASTeX62

A Galactic Plane Deﬁned by the Milky Way H II Region Distribution

L. D. Anderson,1, 2, 3 Trey V. Wenger,4 W. P. Armentrout,1, 3, 5 Dana S. Balser,6 and T. M. Bania7

1Department of Physics and Astronomy, West Virginia University, Morgantown WV 26506, USA 2Adjunct Astronomer at the Green Bank Observatory, P.O. Box 2, Green Bank WV 24944, USA 3Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA 4Astronomy Department, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA 5Green Bank Observatory, P.O. Box 2, Green Bank WV 24944, USA 6National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville VA, 22903-2475, USA 7Institute for Astrophysical Research, Department of Astronomy, Boston University, 725 Commonwealth Ave., Boston MA 02215, USA

ABSTRACT

We develop a framework for a new definition of the Galactic midplane, allowing for tilt (θtilt; rotation ◦ ◦ about Galactic azimuth 90 ), and roll (θroll; rotation about Galactic azimuth 0 ) of the midplane with respect to the current definition. Derivation of the tilt and roll angles also determines the solar height above the midplane. Here we use nebulae from the WISE Catalog of Galactic H II Regions to define the Galactic high-mass star formation (HMSF) midplane. We analyze various subsamples of the WISE catalog and find that all have Galactic latitude scale heights near 0.30 ◦ and z-distribution scale heights near 30 pc. The vertical distribution for small (presumably young) H II regions is narrower than that of larger (presumably old) H II regions (∼ 25 pc versus ∼ 40 pc), implying that the larger regions have migrated further from their birth sites. For all H II region subsamples and for a variety of fitting methodologies, we find that the HMSF midplane is not significantly tilted or rolled with respect to the currently-defined midplane, and therefore the Sun is near to the HMSF midplane. These results are consistent with other studies of HMSF, but are inconsistent with many stellar studies, perhaps due to asymmetries in the stellar distribution near the Sun. Our results are sensitive to latitude restrictions, and also to the completeness of the sample, indicating that similar analyses cannot be done accurately with less complete samples. The midplane framework we develop can be used for any future sample of Galactic objects to redefine the midplane.

Keywords: Galaxy: structure – ISM: H II regions

1. INTRODUCTION the Galaxy have found an asymmetry in the distribu- The midplane, the plane at Galactic latitude b = tion of sources above and below the plane, with more 0 ◦, was defined in 1958 by the IAU subcommission sources found below the IAU plane than above it. This 33b, which set the Galactic coordinate system (Blaauw asymmetry is generally assumed to be the result of the et al. 1960). The IAU midplane definition comes from Sun’s location above the IAU Galactic midplane. the Galactic Center location in B1950 coordinates of Previous studies of the vertical distribution of objects and solar height above the plane can be categorized as arXiv:1812.02244v1 [astro-ph.GA] 5 Dec 2018 (17:42:26.6, −28:55:00) and the north Galactic pole location in B1950 coordinates of (12:49:00, +27:24:00). either using stellar or gas samples. Solar height stud- Ideally, the midplane definition would contain the mini- ies are summarized in Humphreys & Larsen(1995) and mum of the Galactic potential and there would be equal Karim & Mamajek(2017). Studies of stellar samples amounts of material above and below the midplane. The have a long history; perhaps the first such study was vertical distribution of objects with respect to the Galac- done by van Tulder(1942), who found an asymmetry tic midplane tells us fundamental parameters of Galactic in the stellar distribution that implied that the Sun is structure, such as the scale height of the objects studied, 14 ± 2 pc above the plane. Typical stellar studies examine discrepancies in the number of sources toward the Sun’s height above or below the midplane, z , and even the orientation of the midplane itself. Nearly all the north and south Galactic poles to determine the so- previous studies of the vertical distribution of objects in lar height (e.g., Humphreys & Larsen 1995). A typical 2 Anderson et al.

Table 1. Previous Results Analyzing the b and z-Distributions of Galactic Objects

Tracer Galactic Latitude b Height z Scale Heighta Peak Scale Heighta Peak Zone Referenceb (deg.) (deg.) (pc) (pc) ATLASGAL 870 µm cont. 0.3 −0.076 ± 0.008 −60 ◦ < ` < 60 ◦; | b | ≤ 1.5 ◦ 1 ATLASGAL 870 µm cont. 28 ± 2 −6.7 ± 1.1 −60 ◦ < ` < 60 ◦; | b | ≤ 1.5 ◦ 2 ATLASGAL 870 µm cont. 31 ± 3 −4.1 ± 1.7 −60 ◦ < ` < 0 ◦; | b | ≤ 1.5 ◦ 2 ATLASGAL 870 µm cont. 24 ± 1 −10.3 ± 0.5 0 ◦ < ` < 60 ◦; | b | ≤ 1.5 ◦ 2 BGPS 1.1 mm cont. −0.095 ± 0.001 −10 ◦ < ` < 90.5 ◦; | b | < 0.5 ◦ 3 BGPS 1.1 mm cont. 27 ± 1 −9.7 ± 0.6 15 ◦ < ` < 75 ◦; | b | < 0.5 ◦ 4c IR-identified star clusters 0.66 ± 0.07 Entire Galaxy 5 Ultra-compact HII regions Entire Galaxy 6 Ultra-compact HII regions 31 −10 ◦ < ` < 40 ◦; | b | ≤ 0.5 ◦ 7 Ultra-compact HII regions 0.6 30 Entire Galaxy 8 HII regions 42 −11 17.9 < ` < 55.4 ◦; | b | < 1 ◦ 9 d HII regions 39.3 −7.3 Entire Galaxy, RGal < R0 10 e HII regions 33.0 ± 0.06 −7.6 Entire Galaxy, RGal < R0 11 High mass star forming regions 29 ± 0.5 −20 to 0 17.9 < ` < 55.4 ◦; | b | < 1 ◦ 12 ◦ ◦ CH3OH masers 0.4 ± 0.1 −174 < ` < 33 13 HI cold neutral medium ∼ 150 Entire Galaxy 14 CO 0.45 Entire Galaxy 15 Far-IR dust 0.50 Entire Galaxy 16 Far-IR dust 0.32 −0.06 −70 ◦ < ` < 68 ◦; | b | < 1.0 ◦ 17 158 µm [CII] 0.56 73 −28 Entire Galaxy at b = 0 ◦ 18 aAs listed in the paper, regardless of whether value corresponds to the exponential or Gaussian scale height (see Equation1). b1: Beuther et al.(2012); 2: Wienen et al.(2015); 3: Rosolowsky et al.(2010); 4: Ellsworth-Bowers et al.(2015); 5: Mercer et al.(2005); 6: Bronfman et al.(2000); 7: Becker et al.(1994); 8: Wood & Churchwell(1989); 9: Anderson & Bania(2009); 10: Paladini et al.(2004); 11: Bobylev & Bajkova (2016); 12: Urquhart et al.(2011); 13: Walsh et al.(1997); 14: Kalberla(2003); 15: Dame et al.(1987); 16: Beichman et al.(1988); 17: Molinari et al.(2015); 18: Langer et al.(2014) cThese authors correct for the solar offset above the Galactic midplane. Quoted values are our computations from their database, using z = d sin(b) and the Brand(1986) rotation curve. dValues are only for the sample with the most accurate distances. eListed scale height is from the authors’ Gaussian model fit. value for the solar height from stellar studies is 20 pc; The tracers that are most sensitive to HMSF have nar- for example, Ma´ız-Apellániz(2001) used OB stars from row distributions with scale heights ∼ 40 pc, whereas Hipparchos to derive z = 24.2 ± 1.7 pc, Chen et al. the distributions of H I, CO, far-infrared emitting dust, (2001) used stars from an early release of the Sloan Dig- and C II are broader. ital Sky Survey (SDSS) to derive z = 27 ± 4 pc., and The solar height above the plane derived using gas Jurićet al.(2008) found using SDSS data release 3 (with tracers is generally lower than that found from stellar some data release 4) that the z-distributions for stars of tracers (see compilation in Karim & Mamajek 2017). a range of colors and brightnesses are all consistent with Typical values are near 10 pc. For example, Bobylev & z ' 25 pc. Bajkova(2016) found z = 8 ± 2 pc from a sample of We summarize the studies of Galactic latitude and H II regions, masers, and molecular clouds and Paladini z-distributions that use gas tracers in Table1, focus- et al.(2003) found z = 9.3 ± 2 pc using a sample of ing on works that use tracers sensitive to HMSF. This H II regions. table contains the peak and scale height of the distri- Because it was defined using low-resolution data and butions. If the fits were exponential, we list the stated our measurements have since improved significantly, the scale height. If the fits were Gaussians, we list the scale IAU-defined Galactic midplane may need to be revised height h as (Goodman et al. 2014). We now know that Sgr A∗ lies at FWHM ◦ h = , (1) b = −0.046165 (Reid & Brunthaler 2004), which places 2(2 ln 2)0.5 it below the IAU-defined location of the Galactic Center where FWHM is the full width at half maximum of (although by the IAU’s definition Sgr A is at the Galac- the distribution. For a given sample, we do not ex- tic center (Blaauw et al. 1960)). Goodman et al.(2014) pect significant discrepancies between the exponential investigated the implications of this offset and of the and Gaussian scale heights (Bobylev & Bajkova 2016). Sun lying above the midplane using the extremely long There are larger discrepancies between values derived “Nessie” infrared dark cloud (IRDC). Although Nessie using gas tracers compared to those derived using stellar lies below the midplane as it is currently defined, be- tracers. There is, nevertheless, good agreement that the cause of the Sun’s offset and the offset of Sgr A∗ from various distributions peak below the Galactic midplane. Galactic midplane with H II Regions 3

◦ o az=0 b = 0 , they found that Nessie may actually lie in what xˆ' zˆ yˆ they call the “true” midplane, which is tilted by angle Sun 1 θtilt with respect to the IAU midplane definition . While zˆ' yˆ ' suggestive, this study needs to be expanded to a larger xˆ sample of objects in order to make stronger( claimsx',y',z about')=(x ,0,z' ) SgrA* the midplane definition. Tracers of HMSF formation should define the Galactic R0 x⊙ midplane, although it is difficult to create a large, unbi- ased sample of HMSF regions. Most of the gas tracers are related to massive stars, which are born in the most (x,y,z)=(R ,y ,z ) SgrA* o massive molecular clouds in the Galaxy. The high-mass0 SgrA* SgrA* =0 o stars themselves have lifetimes short enough that they az=270o az=90 are unable to travel far from their birthplaces. For example, an O-star with a space velocity of 10 km s−1 can ySgrA* only travel 100 pc out of the midplane in 10 Myr, and only then if its velocity is entirely in thez ˆ direction. az=180o Other tracers of high-mass stars should be similarly re- Figure 1. Face-on geometry of the two coordinate systems stricted to the midplane. used here, as viewed from the Galactic north pole (alongz ˆ0). In Section2, we first develop the methodology needed Angles are exaggerated for clarity. Unprimed coordinates to redefine the Galactic midplane. We apply this are centered on the Sun, whereas primed coordinates are methodology to the WISE Catalog of Galactic H II Re- centered at Sgr A∗. Angles are indicated with green lines gions (Anderson et al. 2014) in Section4, after first and font. Althoughz ˆ is not exactly out of the plane due to characterizing the vertical structure of the Galaxy’s H II the midplane tilt, we ignore this complication when showing region population in Section3. We therefore define the the coordinates. HMSF midplane, determine the tilt and roll angles of 2 the HMSF midplane with respect to the current IAU the north Galactic pole. In the Sun-centered coordi- definition and determine the Sun’s displacement from nates,x ˆ points from the Sun to the (currently-defined) the HMSF midplane. The WISE catalog does not suf- Galactic center,y ˆ points in the direction of Galactic ◦ fer from the same incompleteness and biases of other azimuth az = 90 , andz ˆ points toward the Galac- ∗ studies, and so may be better suited to determining the tic north pole. In the Sgr A -centered coordinate sys- 0 ∗ HMSF midplane than tracers used previously. tem,x ˆ points from Sgr A in the approximate direction of the Sun,y ˆ0 points in the direction of Galactic az- 2. DEFINING THE GALACTIC MIDPLANE imuth az ' 90 ◦, andz ˆ0 points approximately toward Here, we develop the methodology required to define the Galactic north pole. the midplane using a sample of discrete Galactic ob- We show the geometries of the two coordinate systems jects. Although the derived equations are general, we as- in Figures1,2, and3. The modified midplane can be ˆ0 sume in later sections that the midplane passes through tilted by angle θtilt (rotated about x ; Figure2) and ˆ0 Sgr A∗. Future analyses with more data points may be rolled by angle θroll (rotated about y ; Figure3). The able to relax this assumption. modified midplane takes the form

0 2.1. Coordinate systems z = θtilt(x−xSgrA∗ )+θroll(y−ySgrA∗ )+(z−zSgrA∗ ) . (2) To define the Galactic midplane, we need to use two ∗ coordinate systems: the current IAU Galactic coordi- Sgr A is located at (`SgrA∗ , bSgrA∗ ) = nate system centered on the Sun (x, y, and z) and a new (359.944249 ◦, −0.046165 ◦)(Reid & Brunthaler 2004), one centered on Sgr A∗ (x0, y0, z0) using the “modified” which gives non-zero values for ySgrA∗ and zSgrA∗ . As midplane definition. A Galactic azimuth (az) of zero can be seen in Figure2, degrees connects the Sun and the Galactic Center, and azimuth increases clockwise in the plane as viewed from zSgrA∗ = R0 sin bSgrA∗ . (3)

1 We call this rotation the “tilt” angle to be consistent with 2 Technically, az is diﬀerent between the two coordinate sys- previous authors, although by convention it would be called the tems. We use here the azimuth deﬁned in the current coordinate “pitch” angle. system, but only to orient the reader. 4 Anderson et al.

We can use the geometries in Figures1 and3 to deter- when smaller numbers of H II regions are used. We use mine V2.1 of the WISE catalog3, which contains 1813 known R sin ` ∗ 0 SgrA II ySgrA∗ = . (4) H regions that have ionized gas spectroscopic obser- cos θroll vations, 1130 of which have known distances. This H II We derive conversions between these coordinate sys- region sample extends across the entire Galactic disk to tems in AppendixA, with the main result being the Heliocentric distances > 20 kpc and Galactocentric dis- derivation of z0: tances > 18 kpc (Anderson et al. 2015). Since the WISE 00 0 catalog was derived using 6 -resolution 12 µm data, or z = (R0 − x) sin θtilt cos θroll 200 Spitzer data in crowded ﬁelds, and the nominal H II − (ySgrA∗ − y) sin θroll (5) region size is on the order of arcminutes, confusion is + (z − zSgrA∗ ) cos θtilt cos θroll minimal. Therefore, the WISE catalog suﬀers less from blending of distant regions compared with lower resolu- where     tion studies (see Beuther et al. 2012). The catalog also x d cos ` cos b has no latitude restriction, which removes an additional  y  =  d sin ` cos b  . (6)     impediment to the study of the vertical distribution of

z d sin b HMSF (see Section 4.4.3). We can therefore compute z0 for each Galactic object, We compute the height above the plane, z, for each given its (x, y, z) values, the rotation angles, and the WISE catalog H II region using Equation6: location of Sgr A∗. We give the z-heights for locations z = d sin(b) , (8) along az = 0 ◦ in Table2.

2.1.1. Midplane tilt, midplane roll, and the Sun’s height where d is the Heliocentric distance and b latitude from the nominal H II region centroid position in the catalog. The tilt angle, which is apparent in Figure2, does not The definition of z has no correction for the Sun’s height have a compact analytical form unless we make some above the midplane, and so differs from that used in simplifying assumptions. Its complete form can be found some recent studies (e.g., Ellsworth-Bowers et al. 2015). by solving (cf. AppendixA): There is also no correction for the displacement of Sgr A∗ ◦ 0 below b = 0 . z = R0 sin θtilt cos θroll − ySgrA∗ sin θroll If available, the catalog distances are from maser par- +z ∗ cos θ cos θ . SgrA tilt roll allax measurements (e.g., Reid et al. 2009, 2014), but otherwise they are kinematic distances. The original To simplify the equation for θtilt, we assume that θtilt ' ◦ WISE catalog used the Brand & Blitz(1993, hereafter 0 (Goodman et al. 2014, find θtilt ' 0.1 ), so cos θtilt ' B93) rotation curve for kinematic distances. Here, we 1. We can further assume that θroll ' 0 so that update all known H II region distances using the method cos θroll ' 1 and ySgrA∗ sin θroll is small compared to the other terms. The tilt angle is then: of Wenger et al.(2018, hereafer “MC”), which better accounts for uncertainties in kinematic distances. Be- 0 −1 z + zSgrA∗ cause of their large uncertainties, the catalog contains θtilt ' sin (7) ◦ R0 no kinematic distances for H II regions within 10 in Galactic longitude of the Galactic Center, within 20 ◦ of This differs from the angle used in Ellsworth-Bowers the Galactic anti-center, and for any region where the et al.(2013) by the additional term zSgrA∗ . distance uncertainty is > 50%. We use R0 = 8.34 kpc The roll angle is apparent in Figure3. There is no throughout (Reid et al. 2014). compact solution for θroll under reasonable assumptions. Because of the Galactic warp, we cannot use all cata- Its full form can be found by solving Equation A6. loged H II regions to investigate vertical structure in the Galaxy. The warp is known to begin around the solar 3. THE WISE CATALOG OF GALACTIC HII orbit (Clemens et al. 1988), at R ' 8.34 kpc. We in- REGIONS 0 vestigate the warp by plotting the z distribution of H II We wish to investigate the HMSF midplane using regions as a function of RGal in the top panel of Figure4. H II regions in the WISE Catalog of Galactic H II Re- Each point in the top panel of Figure4 represents an H II gions (hereafter the WISE catalog Anderson et al. 2014), region, color-coded by its Galactic longitude. In agree- which contains all known Galactic H II regions. First, however, we characterize this sample of H II regions, and define subsamples to investigate how results may change 3 http://astro.phys.wvu.edu/wise/ Galactic midplane with H II Regions 5

Table 2. Reference points in the two coordinate systems Location Sun-centered Sgr A∗-centered (`, b, d) z z0 0 Sun: (—,—,0) 0 R0 sin θtilt cos θroll − zSgrA∗ cos θtilt cos θroll = z ∗ Sgr A :(`SgrA∗ , bSgrA∗ ,R0) R0 sin bSgrA∗ = zSgrA∗ 0 ◦ ◦ Current GC: (0 , 0 ,R0) 0 −zSgrA∗ cos θtilt cos θroll

zˆ zˆ' xHII Sun x⊙ yˆ xˆ xˆ' bHII b yˆ ' z SgrA* C z' dHII HII o urren ⊙ R =0 t midp 0 lane o zSgrA* (b=0 o zHII' θtilt ) az=180 az=0o SgrA* Modified midplane

xHII' x⊙' ' Figure 2. Geometry used for calculations of the midplanedint tilt rotation angle and the Sun’s height above the plane. Angles are exaggerated for clarity and are indicated with green lines and font. Primed quantities correspond to distances in the coordinate ∗ system defined by the modified midplane that passes through Sgr A . The modified midplane is tilted by an angle θtilt, for a cut through the plane at Galactic azimuth of 0 ◦ as viewed from an azimuth of 270 ◦. One example H II region is shown to represent the analysis done in the next section on the entire H II region population. Althoughy ˆ is not exactly out of the plane due to the fact that `SgrA∗ 6= 0, we ignore this complication when showing the coordinates.

zˆ zˆ' Sun yˆ yˆ' x xˆ' z' ˆ ⊙ y ySgrA* HII zHII o ) =0o (b=0 plane z mid SgrA* rrent Cu zHII' Modified midplane θroll o o az=270 SgrA* az=90

yHII'

Figure 3. Geometry used for calculations of the roll angle. Angles are exaggerated for clarity and are indicated with green lines and font. Primed quantities correspond to distances in the coordinate system defined by the modified midplane that passes ∗ ◦ through Sgr A . We show the geometry for a roll angle θroll, for a cut through the plane at Galactic azimuth of 90 as viewed from an azimuth of 180 ◦ (Sgr A∗ is in the foreground). The third and fourth Galactic quadrants are therefore on the left of the diagram and the first and second quadrants are on the right. One example object, an H II region, is shown. Althoughx ˆ is not exactly out of the plane due to the fact that `SgrA∗ 6= 0, we ignore this complication when showing the coordinates. 6 Anderson et al.

360 ◦ 600 large concentration near ` = 80 are associated with the Cygnus X complex. 400 270 200 3.1. Subsamples of the WISE Catalog 0 180 Due to issues of completeness and biases introduced -200 by large H II region complexes, we deﬁne multiple sub- -400 90 samples of the WISE catalog. We run our analyses on Galactic Longitude (deg.) Height Above Plane (pc) -600 these subsamples to investigate potential biases in our 0 results. We also test the impact of using diﬀerent rota- 600

400 tion curve models. For all subsamples, we include only regions with R < R , with R = 8.34 kpc. 200 Gal 0 0

0 3.1.1. Galactic Quadrants 0 5 10 15 20 Standard Deviation (pc) Galactocentric Radius (kpc) The completeness of the WISE catalog varies across the Galaxy. Nearly all recent H II region surveys have Figure 4. The warping and flaring of the Galactic disk. Shown is the height above the plane relative to IAU defini- taken place in the northern sky, and therefore there are tion, z, versus Galactocentric radius RGal for the H II regions many more known H II regions in the first Galactic quad- in the sample (top panel). The color of each point corre- rant compared to the fourth. The luminosity distribu- sponds to the Galactic longitude of the region. We decrease tion of the first quadrant sample suggests that it is com- the symbol size at low values of RGal for clarity. The warp plete for all H II regions ionized by single O-stars, but begins near the solar circle, RGal ' 8.34 kpc (dashed vertical this is not the case in the fourth quadrant (W. Armen- line), in agreement with previous studies. The warp is toward trout et al., 2018, in prep.). This asymmetry may intro- the North Galactic pole in the first and second quadrants (black/purple/blue points) and toward the South Galactic duce a bias into our analysis. We therefore perform our pole in the third and fourth quadrants (green/yellow/orange analyses below using two Galactic longitude subsamples ◦ ◦ points). In the bottom panel, black filled squares show the that both have RGal < R0: one from 10 < ` < 75 standard deviation of the z-heights derived from Gaussian (hereafter the “first quadrant sample”) and one contain- fits to sources in 1 kpc bins. Within the solar circle, the ing all regions in the first and fourth quadrants (the “in- standard deviation is . 50 pc, and this can be thought of ner Galaxy sample”). The first quadrant sample has 682 as the scale height. Outside the Solar circle, the standard H II regions, 458 of which have known distances, and the deviation increases rapidly due to the Galactic warp. inner Galaxy sample has 1149 H II regions, 613 of which have known distances. ment with previous results, the warp as traced by H II regions begins near the solar circle ( RGal ' 8.34 kpc) 3.1.2. HII Region Complexes and extends toward the north Galactic pole in the first H II regions are frequently found in large complexes Galactic quadrant and toward the south Galactic pole containing many individual H II regions. The WISE cat- in the third Galactic quadrant. alog lists entries for each individual region in the com- The standard deviation of the H II region sample plex and therefore the results of a statistical study will shown in the bottom panel of Figure4 is relatively con- differ based on whether the complex is considered to be stant in the inner Galaxy. The inner Galaxy values are one H II region or many. There are ∼ 600 objects in the all < 50 pc, and this can be thought of as the scale WISE catalog that do not have ionized gas or molec- height. In the outer Galaxy, the standard deviation ular spectroscopic observations, but are placed into a increases with Galactocentric radius as a result of the complex on the basis of the appearance of the complex Galactic warp. This agrees with the results of the H II in mid-infrared and radio continuum data (e.g., W49, region study by Paladini et al.(2004) and the CO study W51, Sgr B2, etc.). The distance to these regions are as- by Malhotra(1994). We exclude RGal bins that have sumed to be that of the other complex members. These fewer than 10 sources from these computations. large complexes may bias our results because there are In the analysis of the HMSF midplane (Section4), we many regions in the catalog at particular Galactic loca- exclude sources with RGal > R0, where R0 = 8.34 kpc. tions. This bias may be warranted because these large We also exclude regions with distance uncertainties > complexes may better define the midplane (as found by 50%. We show the Galactic locations of the H II regions V. Cunningham et al., 2018, in prep.). studied here in Figure5. The lack of regions within 10 ◦ We test for the effect of complexes on our results by of the Galactic Center is in part caused by a lack of running the analyses on two subsamples, one only con- known distances for those regions. Most regions in the taining “unique” H II regions (i.e., each complex contains Galactic midplane with H II Regions 7

2 All 1 Known Dist. Group

-1

Galactic Latitude (deg.) -2 90 60 30 0 330 300 270 Galactic Longitude (deg.)

Figure 5. Galactic distribution of H II regions with RGal < R0. Regions without known distances are shown as black dots, those with kinematic distances are shown with red dots, and the few “group” regions that are in large H II region complexes but which lack individual spectroscopic observations are shown with green dots. Unless they have trigonometric parallax distances or velocities consistent with the nuclear disk, sources within 10 ◦ of the Galactic Center lack known distances. The latitude range here is restricted to show greater detail, and this excludes some regions from the plot. only one catalog entry), and the other that has all re- 400 gions, including “group” regions that that are in large Inner Galaxy First Quadrant H II region complexes but which lack individual spectro- 300 scopic observations. For the group regions, we assume the kinematic distance of the other complex members. 200 We only show results from these subsamples in the ﬁrst Number Galactic quadrant. In the ﬁrst Galactic quadrant, the 100 unique subsample contains 605 H II regions, 408 of which 0 have known distances, and the group subsample contains 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1132 H II regions, 725 of which have known distances. zB93 / zR14 3.1.3. Rotation Curves

400 The majority of the WISE catalog distances are kine- Inner Galaxy matic, and are therefore sensitive to the choice of rota- First Quadrant 300 tion curve. Diﬀerent distances result in diﬀerent values for z (cf. Equation8). We examine how our results 200 change when kinematic distances are computed using

Number the B93 curve, the Reid et al.(2014, hereafter “R14”), 100 and the MC analysis. We do not change the parallax distances in any of our trials. R14 lists multiple 0 rotation curve models; the one we use here has a so- −1 0.90 0.95 1.00 1.05 1.10 1.15 1.20 lar circular angular velocity Ω0 = 235 km s , a solar zMC / zR14 distance from the Galactic Center R0 = 8.34 kpc, and Ω( RGal) = Ω0 − 0.1 RGal. In general, the B93 curve Figure 6. Distribution of z-heights for the B93, R14, and gives larger distances compared to the R14 curve, and MC distances, expressed as a ratio. The B93 curve returns larger Heliocentric distances on average when compared with therefore the B93 z- distances are larger than those of distances from the R14 curve and MC distances, and there- R14 (Figure6, top panel). The MC and R14 curve z- fore on average has larger z heights. There is no significant distances are similar (Figure6, bottom panel). difference between the inner Galaxy and first quadrant samples. The bi-modal distribution in the top panel is due to 3.2. Characterizing the HII Region Vertical B93 distances being preferentially larger than R14 distances Distribution for a given source velocity if the source is nearby, but the We characterize the WISE catalog Galactic latitude two rotation curves giving similar distances otherwise. The distribution for all H II regions in the sample and the z- larger uncertainties zB93/zR14 > 1.05 are all from sources at the near kinematic distance. This, combined with the fact distribution for regions with known distances in Table3. that the distances themselves are not evenly distributed gives These analyses do not rely on the modified midplane def- rise to the bi-modal nature. inition in Section2, but are comparable to those derived by previous authors in Table1. 8 Anderson et al.

80 120 Scale: 0.25±0.01o First Quadrant Scale: 0.29±0.01o Inner Galaxy o o Peak: -0.05±0.01 100 Peak: -0.06±0.01 60 80 40 60

Number Number 40 20 20 0 0 -2 -1 0 1 2 -2 -1 0 1 2 Galactic Latitude (deg.) Galactic Latitude (deg.)

Figure 7. Galactic latitude distributions for H II regions in the first quadrant (left) and the inner Galaxy (right). The KDE is shown with a solid black curve, and this is fit with a Gaussian function shown as the red curve. The Gaussian fits to all subsamples peak at small negative values of b. That these distributions peak at negative latitudes can be explained if the Sun lies above the HMSF midplane. The scale heights from the Gaussian fits are all near 0.30 ◦.

40 60 Scale: 32.34±0.76 pc First Quadrant Scale: 33.21±0.66 pc Inner Galaxy Peak: -6.49±0.76 pc MC 50 Peak: -3.25±0.66 pc MC 30 40 20 30

Number Number 20 10 10 0 0 -200 -100 0 100 200 -200 -100 0 100 200 z (pc) z (pc)

Figure 8. Height above the plane, z, for the first quadrant (left) and inner Galaxy samples (right) for MC distances. The KDE is shown with a solid black curve, and this is fit with a Gaussian function shown as the red curve. The Gaussian fits to the various subsamples peak at small negative values of z. That these distributions peak at negative z-heights can be explained if the Sun lies above the HMSF midplane.

Table 3. H II Region b and z-Distributions Galactic Latitude b Height z Sample Rot. Curve Modiﬁcation Number Scale Height Peak Number Scale Height Peak (deg.) (deg.) (pc) (pc) First Quadrant MC 682 0.27 ± 0.01 −0.05 ± 0.01 458 32.3 ± 0.8 −6.5 ± 0.8 First Quadrant MC Unique 605 0.28 ± 0.01 −0.05 ± 0.01 408 33.5 ± 0.8 −6.2 ± 0.8 First Quadrant MC Group 1132 0.29 ± 0.01 −0.05 ± 0.01 725 32.3 ± 0.8 −6.5 ± 0.8 First Quadrant R14 682 0.27 ± 0.01 −0.05 ± 0.01 475 30.1 ± 0.8 −6.2 ± 0.8 First Quadrant B93 682 0.27 ± 0.01 −0.05 ± 0.01 475 30.8 ± 0.8 −6.5 ± 0.8 Inner Galaxy MC 1149 0.31 ± 0.01 −0.05 ± 0.01 613 33.2 ± 0.7 −3.2 ± 0.7 First Quadrant MC r < 2 pc 130 0.21 ± 0.01 −0.02 ± 0.01 104 24.4 ± 0.7 −4.6 ± 0.7 First Quadrant MC 2 < r < 5 pc 211 0.34 ± 0.01 −0.04 ± 0.01 200 40.3 ± 1.1 −6.0 ± 1.1 First Quadrant MC r > 5 pc 159 0.29 ± 0.01 −0.07 ± 0.01 154 41.2 ± 0.9 −9.7 ± 0.9 Galactic midplane with H II Regions 9

3.2.1. Galactic Latitude Distribution high-mass star forming regions (Urquhart et al. 2011) The distribution of Galactic latitudes is representative (see Table1). of the z-height distribution, but since it does not require 3.2.3. Variations with HII Region Size distances to the objects the analysis can be done on a larger sample of H II regions. Figure7 shows the first Finally, we test how the b- and z-distributions changes quadrant and inner Galaxy H II region Galactic latitude when the sample is restricted to H II regions of various distributions, although we perform the same analysis for physical sizes. The size of an H II region is a proxy for all subsamples defined previously, for all rotation curves. its age (e.g. Spitzer 1978). Diffuse H II regions are dif- We plot the “kernel density estimation” (KDE) in the ficult to detect (Lockman et al. 1996; Anderson et al. black curve. The KDE estimates the underlying dis- 2017), and excluding larger diffuse regions from the sam- tribution, and an analysis of the KDE is free from the ple may have an impact on the derived results. We di- uncertainties associated with the choice of bin size. For vide the first quadrant sample into three physical size this and all subsequent analyses, we use the “Epanech- groups based on the WISE catalog radius r: r < 2 pc, nikov” kernel with the optimal bandwidth as suggested 2 < r < 5 pc, and r > 5 pc. We show these distribu- by Silverman(1986, their Equation 3.31). We fit the tions and fits in Figure9, and give the fit parameters in KDE distributions with Gaussian functions and store Table3. Smaller regions have a smaller scale height of the results in Table3. ∼ 25 pc, whereas the largest regions have scale heights For all subsamples, the peak of the Galactic latitude of ∼ 40 pc. Furthermore, the larger region distributions distribution is slightly below b = 0 ◦ (possibly indicating are consistent with larger solar heights and hence larger 0 tilt angles. Assuming the smaller regions are on average a positive value for the solar height z ), ranging from −0.04 to −0.06 ◦. The scale height, again the standard younger, this result is consistent with migration of older deviation of a Gaussian fit to the Galactic latitude dis- regions out of the plane as they age. tribution, is between 0.25 ◦ and 0.30 ◦ for all subsamples. These scale height values are similar to those found for 4. THE HMSF MIDPLANE DEFINED BY HII other high-mass star tracers (Table1). It is interest- REGIONS ing that our sample of H II regions, which spans a wide We use the results from Section2 to define the HMSF range of evolutionary stages, has the same scale height as midplane with H II regions. In this process, we deter- tracers that are more sensitive to future star formation mine the tilt and roll angles, and also the solar offset (e.g., sub-mm/mm clumps). We can infer that the life- from the midplane. Our analysis necessarily ignores lo- time of H II regions is short enough to not make a large cal deviations (e.g., Malhotra 1994, 1995) to find the av- difference in their b-distribution compared with younger erage midplane definition most consistent with the data. objects. As before, we exclude H II regions with RGal > R0,H II regions within 10 ◦ in Galactic longitude of the Galactic 3.2.2. z-Distribution Center, all regions with distance uncertainties > 50%. For regions with known distances, we can study the ◦ z-height distribution directly. The z-distributions, for 4.1. Midplane Tilt with θroll = 0 which we show examples in Figure8, are approxi- We assume that the H II region z0 distribution will 0 mately Gaussian for all subsamples. There are, how- peak at 0 pc for the “correct” value of z . Changing 0 0 ever, “wings” to the distributions at high and low val- z alters z for each H II region in the sample (Equa- 0 ues of z. As with the Galactic latitude distributions, tion5). This value of z also results in a unique value we fit the KDEs of the z distributions with Gaussian for θtilt (Equation7). With the limited number of fourth functions and store the results in Table3. The anal- quadrant H II regions with known distances, θroll is diffi- ysis of the z-distribution is necessarily limited to only cult to constrain, and is often not acounted for in other H II regions with known distances, which is a smaller analyses of the midplane (e.g., Goodman et al. 2014). 0 subsample compared to that used in the Galactic lat- We vary z from −30 pc to 40 pc in steps of 0.5 pc itude analysis. All subsamples peak at small negative and recompute z0 for all H II regions in the sample. For 0 0 values, from −3 to −6 pc, again implying that the Sun each distribution of z values (for a given value of z ), is located above the midplane. The scale heights for the we fit a Gaussian function to the KDE to determine the various subsamples range from 30 to 34 pc. All distance peak of the distribution (as in Figure8). We show the 0 methods return similar results. These values are similar values of the fitted Gaussian peaks as a function of z 0 to those found for ultra-compact H II regions (Bronfman and θtilt in Figure 10 and give the derived values of z et al. 2000), sub-mm clumps (Wienen et al. 2015) and and θtilt in Table4. 10 Anderson et al.

Table 4. HMSF Midplane Parameters

◦ θroll = 0 θroll Free 0 0 Sample Rot. Curve Modiﬁcation Number z θtilt z θtilt θroll (pc) (deg.)(pc)(deg.)(deg.) First Quadrant MC 458 5.5 ± 2.5 −0.01 ± 0.01 5.6 ± 0.7 −0.01 ± 0.01 0.08 ± 0.01 First Quadrant MC Unique 408 4.5 ± 2.5 −0.02 ± 0.01 6.5 ± 0.8 0.00 ± 0.01 0.10 ± 0.01 First Quadrant MC Group 725 5.5 ± 2.5 −0.01 ± 0.01 9.4 ± 0.6 0.02 ± 0.01 0.11 ± 0.01 First Quadrant R14 475 5.0 ± 2.5 −0.01 ± 0.01 4.2 ± 1.1 −0.02 ± 0.01 0.03 ± 0.01 First Quadrant B93 475 5.5 ± 2.5 −0.01 ± 0.01 5.7 ± 1.2 −0.01 ± 0.01 0.04 ± 0.01 Inner Galaxy MC 613 −2.8 ± 1.6 −0.07 ± 0.01 1.3 ± 0.4 −0.04 ± 0.01 0.04 ± 0.01

First Quadrant MC dcut = 4.7 kpc, α = −0.66 306 4.8 ± 3.1 −0.01 ± 0.02 ········· First Quadrant MC |b| < 0.5 ◦ 385 3.5 ± 2.5 −0.02 ± 0.02 15.0 ± 0.6 0.06 ± 0.01 0.05 ± 0.01 First Quadrant MC |b| < 1.0 ◦ 441 6.2 ± 1.6 0.00 ± 0.02 14.3 ± 0.7 0.05 ± 0.01 0.09 ± 0.01 First Quadrant MC r < 2 pc 104 0.8 ± 1.6 −0.04 ± 0.01 −3.3 ± 1.3 −0.07 ± 0.01 −0.04 ± 0.01 First Quadrant MC 2 < r < 5 pc 200 3.5 ± 3.0 −0.02 ± 0.02 18.0 ± 1.2 0.08 ± 0.01 0.15 ± 0.01 First Quadrant MC r > 5 pc 154 18.5 ± 3.5 0.08 ± 0.03 −8.6 ± 1.5 −0.11 ± 0.01 0.02 ± 0.01

Our analysis ﬁnds that the Sun lies ∼ 5.6 ± 2.6 pc 20 Scale: 24.40±0.70 pc First Quadrant Peak: -5.47±0.70 pc MC above the HMSF midplane for the ﬁrst quadrant sample, ◦ 15 r < 2 pc for tilt angles of ∼ −0.01 , and 4.4 ± 1.9 pc below the plane for the inner Galaxy sample, for tilt angles of ∼ 10 ◦ 0 −0.07 . The uncertainties in the derived values of z

Number and θtilt come from allowing the peak to fall within the 5 range ±1 pc. 0 -200 -100 0 100 200 4.2. Midplane Tilt and Roll z (pc) Similar to our investigation of the midplane tilt, we 20 Scale: 41.22±1.10 pc First Quadrant Peak: -5.72±1.10 pc MC can test for the midplane roll by ﬁtting the distributions 15 2 < r < 5 pc of z0 for the H II region sample. For a one-dimensional 0 ﬁt, we cannot allow θtilt (or, equivalently, z ) and θroll 10 to both be free parameters. Instead, we set θroll to a

Number range of discrete values, and fit for θtilt. 5 0 We compute a grid of z distributions for θroll values ◦ ◦ 0 from −0.8 to 0.2 in increments of 0.05 . We then vary 0 -200 -100 0 100 200 z (and hence θtilt) as in Section 4.1 until we find the z (pc) 0 combination of θtilt and θroll where the z distribution of 15 Scale: 40.30±0.97 pc First Quadrant H II regions peaks at 0 pc. We show the combinations of Peak: -9.70±0.97 pc MC r > 5 pc θtilt and θroll that together lead to distributions peaking 10 at z0 = 0 in Figure 11 for the first quadrant and inner Galaxy samples. There is a negative correlation between

Number 5 θtilt and θroll, such that increasing either angle has the same eﬀect on the H II region distribution. From this one-dimensional analysis we cannot determine the best 0 -200 -100 0 100 200 combination of the two angles, but we can constrain one z (pc) angle given the other one.

Figure 9. The z-distributions for small (r < 2 pc; top), 4.3. Three-Dimensional Fit for Midplane Tilt and Roll medium (2 < r < 5 pc; middle), and large (r > 5 pc; bottom) first quadrant H II region samples. The smallest re- Instead of testing discrete values of θtilt and θroll by 0 gions have the narrowest distribution. The black lines are fitting the one-dimensional distribution of z for H II the KDEs and the red lines are Gaussian fits to the KDEs. regions, we can also simply fit a plane of the form in Equation2 to the H II region distribution. In this way, we simultaneously fit for the Sun’s height and the two midplane rotation angles. The downsides to this Galactic midplane with H II Regions 11 θ θ tilt (deg.) tilt (deg.) -0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2 15 20 First Quadrant Inner Galaxy θ o θ o 10 roll = 0.00 15 roll = 0.00 5 10 0 5 Distribution (pc) Distribution (pc) ’ -5 ’ 0 -10 -5

Peak of z -15 Peak of z -10 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 Sun’s Height Above Plane (pc) Sun’s Height Above Plane (pc)

0 Figure 10. Peak of the H II region distribution as a function of solar height above the plane, z for the ﬁrst quadrant (left) 0 and inner Galaxy (right) samples. The intersection of the blue shaded region with the x-axis shows values of z that result in H II region distributions that peak from −1 to 1 pc.

0.2 First Quadrant 0.2 Inner Galaxy 1D Fit 1D Fit 0.0 3D Fit 0.0 3D Fit

-0.2 -0.2 (deg.) (deg.) roll roll

θ -0.4 θ -0.4

-0.6 -0.6

-0.2 0.0 0.2 0.4 0.6 0.8 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 θ θ tilt (deg.) tilt (deg.)

Figure 11. Variations in θroll as a function of θtilt for the first quadrant (left) and inner Galaxy (right) samples. Each point corresponds to the two tilt angles where the z0-distribution of H II regions is centered at zero. There is a negative correlation between the two angles. Blue squares show the values of θtilt we derived from one-dimensional fits with θroll = 0, whereas the red circles indicate results from three-dimensional fits explained later in Section 4.3. Error bars are 3σ. method are that it is difficult to compare with the one- roll angle is generally small as well, and almost always ◦ dimensional fits frequently done by previous authors, positive; θroll = 0.08 for the first quadrant sample and ◦ and may produce results inconsistent with those of the θroll −0.04 for the inner Galaxy sample. As can be seen one-dimensional fits. in Figure 11, the values of θtilt and θroll derived from We fit the plane using a robust least-squares mini- the three-dimensional fits are broadly consistent (∼ 5σ) mization routine. The robust least squares fit reduces with the one-dimensional relationships. This gives us the impact of outliers on the fit results by minimizing the additional confidence in the derived values. “loss function,” ρ(z), where z is the squared residuals. √ 4.4. Effects of Completeness and Latitude Restrictions We use a “soft l1” loss function, ρ(z) = 1 + z2 − 1,, which is similar to the “Huber” loss function. As be- Since the WISE catalog contains a greater quantity of fore, we force the plane to pass through the location of extremely distant sources compared with other catalogs Sgr A∗. We perform these three-dimensional fits for the of star formation regions, we here examine if our results same subsamples as the one-dimensional fits and store would change if the sample were less complete. the results in the final three columns of Table4. 4.4.1. Artificial Distance Cutoff The results from the three-dimensional fits are similar to those found previously in the one-dimensional fits, de- As a first test, we restrict our first quadrant sample spite allowing for a second midplane tilt angle. We find by removing sources above a range of Heliocentric dis- small absolute tilt angles and small positive values for tances from 1.75 to 8.25 kpc. We then fit the KDE dis- the Solar height about the plane. In the first quadrant tributions of these restricted samples with Gaussians as ◦ 0 before, and show the results in Figure 12. We see in this sample, for example, θtilt = −0.01 and z = 5.7. The figure that the peak of the z distributions varies from 12 Anderson et al.

20 100 100 z Peak CDF 80 80 10

60 60 0

40 CDF (%) 40 CDF (%) z Peak (pc)

-10 20 BGPS 20 HII Modified HII -20 0 0

0 2 4 6 8 150 BGPS Heliocentric Distance Cutoff (kpc) HII Modified HII Figure 12. Variations in the peak of H II region distributions restricted by Heliocentric distance cutoﬀs from 1.75 to 100 8.25 pc (solid curve). Also shown is the cumulative distribution function for the H II region sample unrestricted by Number distance (dotted curve). Imposing a distance cutoﬀ changes 50 the derived z-distribution peak from −20 to +15 pc.

−19 pc to +17 pc. This simple analysis shows that an 0 artificial distance cutoff may have a significant effect on 0 5 10 15 20 the derived values. Heliocentric Distance (kpc)

4.4.2. Malmquist Bias Figure 13. First-quadrant distributions of Heliocentric distances for the BGPS and H II region samples. Since the A flux-limited distribution of course does not have a ◦ hard distance cutoff. We attempt to model a more ac- BGPS catalog is limited to within | b | < 0.5 in the first quadrant, we also show the H II region distribution with this curate Malmquist bias by comparing the H II region dis- limitation. The BGPS distribution is heavily weighted to- tance distribution to that of another flux-limited sam- ward relatively nearby sources compared to the H II region ple. We choose to do the comparison with the Bolo- distribution. cam Galactic Plane Survey (BGPS) catalog (Ellsworth- Bowers et al. 2015), although any flux-limited sample that a source with a Heliocentric distance d is kept in could serve the same purpose. the catalog is therefore: The BGPS catalog contains 3508 mm clumps, 1710 of  which have kinematic distances. In the first quadrant 1, if d < dcut ◦ ◦ p = (9) zone 10 < ` < 75 , there are 2843 clumps identified −α (d − dcut) , if d ≥ dcut . from mm-wave observations, 1214 of which have kinematic distances. The distribution of BGPS Heliocentric We iterate dcut and α in the respective ranges 0 to 12 kpc distances differs from that of the WISE H II regions in and 0 to 1 to create modified H II region distributions, that it has a stronger peak near 5 kpc and a steeper de- and perform a Kolmogorov-Smirnov (K-S) test on the crease thereafter (see below). This indicates that there is BGPS and modified H II region distributions. The K-S either an asymmetry in the mm-clump/H II region ratio test can determine the likelihood that two samples are or that there is Malmquist bias in the subset of BGPS drawn from the same parent distribution. sources with known distances. We find that the two distributions are most similar We attempt to evaluate the impact of potential when α = −0.66 and dcut = 4.7 kpc. We show the Malmquist bias by applying a source removal function Heliocentric distance cumulative distribution functions to the H II region sample that more closely approximates (CDFs) before and after the modification in Figure 13, the effects of Malmquist bias. We create a modified H II top panel, and the distributions themselves in Figure 13, region distribution by keeping all sources in the catalog bottom panel. that have Heliocentric distances less than a cutoff dis- Fitting this modified H II region distribution using tance dcut. For sources with distances greater than the one-dimensional fits as in Section3 does impact the cutoff value, we apply a power law source removal func- derived parameters, although the fits to the modified tion with a power law index α. The percent likelihood distribution give similar peaks and scale heights as are Galactic midplane with H II Regions 13 found for the complete distribution. Repeating the of ∼ 30 pc. These values are dependent on the size of Sun’s height analysis, we find that the modified H II re- the H II regions themselves. The smallest H II regions gion distribution is consistent with the Sun lying 12.5 pc (< 2 pc radius) have the smallest scale height distribu- ◦ above the midplane, for a tilt angle θtilt = 0.04 . These tion of 26 pc. Larger H II regions have a scale height of values are considerably larger than the unmodified val- ∼ 40 pc. Since the size of an H II region depends on its ◦ ues of 5.5 pc and θtilt = −0.01 . We conclude that age, these results may indicate a broadening of the H II Malmquist bias can significantly alter the derived val- region distribution as the regions themselves evolve. ues of the Galactic midplane. We found that the HMSF midplane is not significantly different from the current IAU midplane, and that the 4.4.3. Latitude Restrictions Sun is near to the HMSF midplane. Values for the Most surveys of the Galactic plane are restricted in first quadrant and inner Galaxy samples are similar, al- latitude. For example, the BGPS was limited to | b | < though the first quadrant sample analysis supports a so- ◦ 0.5 . When creating the WISE catalog, we searched lar height of ∼ 5 pc above the current midplane and the ◦ WISE data within 8 of the plane, but included known inner Galaxy sample analysis supports a solar height of a regions outside this range. The WISE catalog therefore few pc below the current midplane. The tilt and roll an- is more complete in latitude compared with catalogs de- gles as defined here are negatively correlated, although rived from most other Galactic plane surveys. To test when θroll is a free parameter we find similar values for the effect of this latitude limitation, we restrict our first- θtilt and the solar height as when θroll is set to zero. We ◦ ◦ quadrant sample to within | b | < 0.5 and | b | < 1.0 caution that the roll angle is not well-constrained due to and repeat the above analyses. We again find the peak a lack of H II regions with known distances in the fourth and scale height are essentially unchanged. The solar Galactic quadrant. ◦ ◦ height is 3.0 pc for | b | < 0.5 and 6.5 pc for | b | < 1.0 , Our values for the solar height are ∼ 15 pc less than ◦ ◦ for tilt angles θtilt = −0.03 and θtilt = 0.00 , respec- those found in studies of stars, but they are consistent tively. The former values are considerably different from with many results of HMSF tracers. The meaning of this ◦ the unrestricted values of 5.8 pc and θtilt = −0.01 . discrepancy is unclear. The stellar samples are compiled ◦ Therefore, restricting the sample to within | b | < 0.5 over a different portion of the Galaxy compared to ours, significantly changes our results, but there is no such ef- since extinction drastically limits the distance stars can ◦ fect if limited to | b | < 1.0 . We conclude that surveys be seen in the midplane. The discrepancy between our ◦ with | b | < 1.0 can reproduce the z-distribution re- results may indicate that near to the Sun there is a large- sults from our more complete sample, but surveys with scale displacement in the stellar population. Since many ◦ | b | < 0.5 cannot. of the stellar studies rely on counting stars toward the 5. SUMMARY north and south Galactic poles, asymmetries in the stellar distribution would alter the derived result for the We developed a framework for studies of the Galactic solar height. For example, Xu et al.(2015) discovered midplane, assuming that the midplane passes through an asymmetry in the main-sequence star counts using the location of Sgr A∗. We allowed for rotation of the data from the Sloan Digital Sky Survey such that there midplane about Galactic azimuths of 90 ◦ (the “tilt”; are more stars in the north 2 kpc from the Sun, more θ ) and 0 ◦ (the “roll”; θ ). Our framework can be tilt roll stars in the south between 4 − 6 kpc from the Sun, and applied to any sample of Galactic objects to determine more stars in the north between 8−10 kpc from the Sun. the midplane, thereby determining the tilt and roll an- Such asymmetries may make determinations of the solar gles with respect to the current midplane definition and height difficult using star counts. also the Sun’s height above the midplane. Our values are, however, broadly consistent with the We applied this framework to the WISE Catalog of results of Wegg et al.(2015). They found from near- Galactic H II Regions to define the high-mass star for- infrared star counts of red clump giants that the mean mation (HMSF) midplane. In other work (Armentrout latitude in the Galactic long bar is b ' −0.1 ◦ and that et al., 2018, in prep.), we have found that the WISE cat- the long bar lies in the midplane after accounting for alog is statistically complete for all first-quadrant H II the midplane tilt. At a longitude of the end of the long regions ionized by single O-stars, giving us a volume- bar of ` ' 30 ◦, a distance of 6000 kpc, b = −0.10 ◦, limited sample. The fourth quadrant sample is less com- θ = −0.01 ◦, and θ = 0 ◦, z0 ' −4 pc. If we redo plete, and we therefore analyze the first quadrant and tilt roll the calculation with the first-quadrant sample values of inner Galaxy (first and fourth quadrants) samples inde- ◦ ◦ 0 θtilt = −0.01 and θroll = 0.08 , z ' −0.1 pc, which pendently. We computed a Galactic latitude H II region implies that the long bar is exactly in the modified mid- scale height of ∼ 0.30 ◦, and a z-distribution scale height 14 Anderson et al.

◦ plane. Using the inner Galaxy values of θtilt = −0.04 ◦ 0 and θroll = 0.04 , z ' −4 pc. We tested the robustness of these results and appli- cability of our methodology using various permutations of our sample. Since most Galactic plane surveys are restricted in latitude, we examine H II region samples restricted to within | b | < 1.0 ◦ and | b | < 0.5 ◦. We found that the | b | < 0.5 ◦ sample results are considerably different, indicating that similar studies of the Milky Way vertical distribution should ideally not include a latitude restriction, and certainly cannot be limited to | b | < 0.5 ◦. Introducing an artificial Malmquist bias also changes the results significantly.

This work is supported by NSF grant AST1516021 to LDA. TMB acknowledges support from NSF grant AST 1714688. Support for TVW was provided by the NSF through the Grote Reber Fellowship Program administered by Associated Universities, Inc./National Radio Astronomy Observatory. We thank Bob Ben- jamin for enlightening discussions on early drafts of this manuscript. We thank Virginia Cunningham for help on early analyses for this project. The Green Bank Obser- vatory and the National Radio Astronomy Observatory are facilities of the National Science Foundation oper- ated under cooperative agreement by Associated Uni- versities, Inc. Galactic midplane with H II Regions 15

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APPENDIX A. COORDINATE CONVERSIONS Ellsworth-Bowers et al.(2013) derived the relationship between Galactic positions measured from the current coordinate system that uses the IAU standards and the values measured from the modified Galactic plane. Their derived relationship did not consider the offset of Sgr A∗from (`, b) = (0 ◦, 0 ◦) or the roll of the plane, and we update their calculations with these changes. In our nomenclature, unprimed coordinates refer to the values in the currently-defined Sun-centered coordinate system and primed values refer to the Sgr A∗-centered values that include the tilt and roll of the midplane. See Figures1,2, and3 for images of the various distances and angles. For local Cartesian coordinates with the Sun at the origin, the current IAU definition gives:     x d cos ` cos b  y  =  d sin ` cos b  , (A1)    

z d sin b wherex ˆ points from the Sun to the Galactic Center,y ˆ is in the Galactic plane and points toward a Galactic azimuth of 90 ◦, andz ˆ points toward the north Galactic pole. ∗ ◦ The location of Sgr A is (`SgrA∗ , bSgrA∗ ) = (359.944249, −0.046165 )(Reid & Brunthaler 2004), so

ySgrA∗ = R0 sin `SgrA∗ . (A2) and zSgrA∗ = R0 sin bSgrA∗ . (A3) The tilt angle between the currently-deﬁned Galactic midplane and the modiﬁed plane is: 0 −1 z + zSgrA∗ θtilt = sin . (A4) R0

This diﬀers from the angle used in Ellsworth-Bowers et al.(2013) by the additional term zSgrA∗ . Note that zSgrA∗ is negative for bSgrA∗ < 0. The full translational and rotational matrices resulting from (1) rotating about thez ˆ-axis so thatx ˆ points toward ∗ ◦ the Sun, (2) moving the origin to be centered on Sgr A , (3) rotating about thez ˆ-axis by the angle 360 − `SgrA∗ (or, 0 equivalently, clockwise by `SgrA∗ ) so thatx ˆ points toward the Sun, (4) rotating the Galactic plane about they ˆ-axis (with the Sun at the origin) clockwise by the angle θtilt, and (5) rotating the Galactic plane counterclockwise about thex ˆ-axis by the angle θroll:

 0      x 1 0 0 0 cos θtilt 0 − sin θtilt 0 y0  0 cos θ sin θ 0  0 1 0 0   =  roll roll     0      z  0 − sin θroll cos θroll 0  sin θtilt 0 cos θtilt 0 1 0 0 0 1 0 0 0 1         cos `SgrA∗ sin `SgrA∗ 0 0 1 0 0 R0 −1 0 0 0 x

 ∗ ∗   ∗      − sin `SgrA cos `SgrA 0 0 0 1 0 ySgrA   0 −1 0 0 y          .  0 0 1 0 0 0 1 −zSgrA∗   0 0 1 0 z  0 0 0 1 0 0 0 1 0 0 0 1 1 Although the tilt rotation is defined in reference to the current midplane definition (a rotation centered on the Sun), the roll is defined from the tilted modified midplane (a rotation centered on Sgr A∗). The derived coordinates are therefore:  0    x (R0 − x) cos `SgrA∗ cos θtilt + (ySgrA∗ − y) sin `SgrA∗ cos θtilt − (z − zSgrA∗ ) sin θtilt 0  y   (R − x)(sin ` ∗ cos θ + cos ` ∗ sin θ sin θ ) − (y ∗ − y)(cos ` ∗ cos θ −     0 SgrA roll SgrA tilt roll SgrA SgrA roll        =  sin θ`SgrA∗ sintilt sinroll) + (z − zSgrA∗ ) cos θtilt sin θroll  .  0     z   (R0 − x)(sin `SgrA∗ sin θroll − cos `SgrA∗ sin θtilt cos θroll) + (ySgrA∗ − y)(sin `SgrA∗ cos θroll sin θtilt + cos `SgrA∗ sin θroll)+  (z − zSgrA∗ ) cos θtilt cos θroll (A5) Galactic midplane with H II Regions 17

Since sin(`SgrA∗ ) = −0.0009730 ' 0 and cos(`SgrA∗ ) = 0.9999995 ' 1, these reduce to

 0    x (R0 − x) cos θtilt − (z − zSgrA∗ ) sin θtilt  0  =  ∗ ∗  . (A6)  y   (R0 − x) sin θtilt sin θroll + (ySgrA − y) cos θroll + (z − zSgrA ) cos θtilt sin θroll  0 z (R0 − x) sin θtilt cos θroll − (ySgrA∗ − y) sin θroll + (z − zSgrA∗ ) cos θtilt cos θroll

Since x and y are of order ∼ kpc and because θtilt and θroll are both small, rotations of the midplane will have a 0 0 0 larger fractional aﬀect on derived values in z compared to x and y . In the limit that θroll = 0,

 0    x (R0 − x) cos θtilt − (z − zSgrA∗ ) sin θtilt  0  =  ∗  . (A7)  y   ySgrA − y  0 z (R0 − x) sin θtilt + (z − zSgrA∗ ) cos θtilt

These expressions diﬀer from those in Ellsworth-Bowers et al.(2013) by the additional ySgrA∗ and zSgrA∗ terms.